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  1. #1
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    Calculus Question

    Population grows according to the equation , where k is a constant and t is measured in years. If the population doubles every 10 years, then the value of k is


    (a) 0.069 (b) 0.200 (c) 0.301 (d) 3.322 (e) 5.000
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  2. #2
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    Quote Originally Posted by frozenflames
    Population grows according to the equation , where k is a constant and t is measured in years. If the population doubles every 10 years, then the value of k is

    (a) 0.069 (b) 0.200 (c) 0.301 (d) 3.322 (e) 5.000
    Hello,

    the growth of this population can be described by the equation:
    p=p_{0} \cdot 10^{k\cdot t}

    where p is the actual amount, p_{0} is the starting value.

    Put in the values you know:
    2 \cdot p_{0}=p_{0} \cdot 10^{k\cdot 10}
    2 = 10^{k\cdot 10}
    solve for k:
    log(2) = k\cdot 10
    now you have to use a caculator or a logarthmic table.

    You'll get k\approx .030103
    that means that none of the given results seem to be right.

    Greetings

    EB
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  3. #3
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    Greetings EB:

    I solved for k in the equation P(t) = P_0 e^kt which yields k = 0.1*ln(2) approx= 0.069. As it happens, 0.069 is indeed one of the indicated choices.

    Enjoy the day,

    Rich B.
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  4. #4
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    Quote Originally Posted by Rich B.
    Greetings EB:

    I solved for k in the equation P(t) = P_0 e^kt which yields k = 0.1*ln(2) approx= 0.069. As it happens, 0.069 is indeed one of the indicated choices.

    Enjoy the day,

    Rich B.
    Hello, RichB.

    of course you are right - BUT. when frozenflames referred to the equation I was not aware, that the equation was meant which you used. It all depends on the base you choose for solving this problem. I have choosen for 10 and thats why I came up with a "wrong" result.

    Have a nice day too.

    EB
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  5. #5
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    Quote Originally Posted by earboth
    Hello, RichB.

    of course you are right - BUT. when frozenflames referred to the equation I was not aware, that the equation was meant which you used. It all depends on the base you choose for solving this problem. I have choosen for 10 and thats why I came up with a "wrong" result.

    Have a nice day too.

    EB
    In fact it does not matter (in principle) what you take for b
    (as long as it is positive ) in a growth equation of the form:

    <br />
p(t)=p(0)b^{kt}<br />

    for:

    <br />
p(t)=p(0) c^{\log_c(b)\ kt}<br />

    Here there is a good case for using 2 since we are discussing doubling
    times, so here it might be natural to use:

    <br />
p(t)=p(0)2^{kt}<br />

    RonL
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  6. #6
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    Hi Ron:

    Right you are. I was just fishing for a base that would yield a parameter, k, that appears in the multiple choice list.

    Enjoy.

    Rich
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